Independent quantum systems and the associativity of the product of quantum observables

Main Article Content

Klaus Fredenhagen


We start from the assumption that the real valued observables of a quantum system form a Jordan algebra which is equipped with a compatible Lie product characterizing infinitesimal symmetries, and ask whether two such systems can be considered as independent subsystems of a larger system. We show that this is possible if and only if the associator of the Jordan product is a fixed multiple of the associator of the Lie product. In this case it is known that the two products can be combined to an associative product in the Jordan algebra or its complexification, depending on the sign of the multiple.

Article Details

Emergence of the Classical


Arodź, H., 2019. Ehrenfest’s Theorem Revisited. Philosophical Problems in Science (Zagadnienia Filozoficzne w Nauce), (66), 39–45.

Braun, H. and Koecher, M., 1966. Jordan-Algebren, Grundlehren der mathematischen Wissenschaften. Berlin: Springer-Verlag.

Grgin, E. and Petersen, A., 1976. Algebraic implications of composability of physical systems. Communications in Mathematical Physics [Online], 50(2), pp.177–188. Available at: [Accessed 2 July 2019].

Hanche-Olsen, H. and Střrmer, E., 1984. Jordan operator algebras, Studies in Mathematics 21. Boston, London, Melbourne: Pitman.

Hanche-Olsen, H., 1985. JB-algebras with tensor products are C*-algebras. In: Araki, H., Moore, C.C., Stratila, S.-V. and Voiculescu, D.-V. eds. Operator Algebras and their Connections with Topology and Ergodic Theory, Lecture Notes in Mathematics 1132. Berlin, Heidelberg: Springer, pp.223–229.

Jordan, P., Neumann, J.v. and Wigner, E., 1934. On an algebraic generalization of the quantum mechanical formalism. Annals of Mathematics [Online], 35(1), pp.29–64. Available at: [Accessed 2 July 2019].

Kapustin, A., 2013. Is quantum mechanics exact? Journal of Mathematical Physics, 54(6), p.062107. Available at:

Landsman, N.P., 1998. Mathematical Topics Between Classical and Quantum Mechanics, Springer Monographs in Mathematics. New York: Springer-Verlag.

McCrimmon, K., 2004. A Taste of Jordan Algebras, Universitext. New York: Springer-Verlag.

Moldoveanu, F., 2015. Derivation of Quantum Mechanics algebraic structure from invariance of the laws of Nature under system composition and Leibniz identity. arXiv: 1505.05577 [quant-ph]. [Accessed 20 March 2019].